Problem 41 - Algebra 2 and Trigonometry

Question

In \(39-42,\) solve and check each equation. $$ 4 a+\sqrt{6}=a+\sqrt{96} $$

Step-by-Step Solution

Answer

The solution is \(a = \sqrt{6}\).

Step 1: Isolate the variable term

The given equation is \(4a + \sqrt{6} = a + \sqrt{96}\). Start by isolating the variable term. Subtract \(a\) from both sides to move all terms involving \(a\) to one side. This results in \(3a + \sqrt{6} = \sqrt{96}\).

Step 2: Eliminate the constant term on the left

Now, subtract \(\sqrt{6}\) from both sides to isolate the term with \(a\) on the left: \(3a = \sqrt{96} - \sqrt{6}\).

Step 3: Simplify square roots

Simplify the square roots if possible. Note that \(\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}\). Thus, the equation becomes \(3a = 4\sqrt{6} - \sqrt{6}\).

Step 4: Combine like terms

Combine the like terms on the right side: \(4\sqrt{6} - \sqrt{6} = 3\sqrt{6}\). The equation simplifies to \(3a = 3\sqrt{6}\).

Step 5: Solve for \(a\)

Divide both sides by 3 to solve for \(a\): \(a = \sqrt{6}\).

Step 6: Check the solution

Substitute \(a = \sqrt{6}\) back into the original equation to verify: \(4(\sqrt{6}) + \sqrt{6} = \sqrt{6} + \sqrt{96}\). Calculate the left side: \(4\sqrt{6} + \sqrt{6} = 5\sqrt{6}\). Calculate the right side: \(\sqrt{6} + 4\sqrt{6} = 5\sqrt{6}\). Both sides are equal, confirming that \(a = \sqrt{6}\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is like unwrapping a present. Our goal is to "uncover" or find the value of the unknown variable that makes both sides of the equation equal. Think of the equation as a balance scale. Whatever you do to one side, you have to do to the other to keep it balanced.

In the equation presented, we start with both sides having both constants and terms with the variable. One approach is to move all terms with variables to one side and constants to the other. This helps us see the "core" of the equation more clearly. The basic tools we use in solving equations include addition, subtraction, multiplication, and division—operations that help us rearrange equation components without changing their equality.

It’s often helpful to follow a strategy or sequence of steps. First, simplify each side of the equation by combining like terms. After that, eliminate any constants to isolate the terms with the variable on one side. Finally, solve for the variable using basic arithmetic. This systematic approach ensures you don’t miss a step.

Square Roots

Square roots can seem tricky, but they're all about finding a value that, when multiplied by itself, gives the original number. In our exercise, we encounter square roots on both sides of the equation. Recognizing and simplifying square roots is crucial in solving such equations.

For example, the square root of 96 can be broken down into simpler parts:

• Start by factoring 96 into 16 and 6, because 16 is a perfect square.
• This gives us \((\sqrt{16} \cdot \sqrt{6})\).
• Knowing \((\sqrt{16} = 4)\), we simplify this to \((4\sqrt{6})\).

When you encounter square roots, check if you can simplify them by identifying perfect squares. Simplifying makes equations easier to work with and often clearer to solve. Remember, when dealing with square roots, try to simplify without altering the balance of the initial equation.

Isolating Variable

Isolating the variable is a crucial step in solving any algebraic equation. The aim is to get the variable by itself on one side of the equation. This helps to clearly see the solution.

In our problem, we moved all terms with the variable to one side by subtraction. Once the variable term was alone, it became more straightforward to solve for it.

To isolate a variable, we need to eliminate any coefficients or constants that are attached to it. This often involves:

• Using addition or subtraction to clear terms that are on the same side as the variable but not part of it.
• Dividing or multiplying to get rid of any coefficients in front of the variable.

The ultimate goal is to have the variable equal some number or expression, which represents the solution. Isolating variables simplifies complex equations into ones that are much easier to handle.